Kurata, Hiroshi; Kariya, Takeaki Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model. (English) Zbl 0868.62060 Ann. Stat. 24, No. 4, 1547-1559 (1996). Summary: In a general normal regression model, this paper first derives the least upper bound (LUB) for the covariance matrix of a generalized least squares estimator (GLSE) relative to the covariance matrix of the Gauss-Markov estimator. Second the result is applied to the (unrestricted) Zellner estimator [A. Zellner, J. Am. Stat. Assoc. 58, 977-992 (1963; Zbl 0129.11203)] in an \(N\)-equation seemingly unrelated regression (SUR) model and to the GLSE in a heteroscedastic model. Cited in 7 Documents MSC: 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) Keywords:seemingly unrelated regression model; nonlinear Gauss-Markov theorem; efficiency; Kantorovich inequality; general normal regression model; least upper bound; covariance matrix; generalized least squares estimator; Gauss-Markov estimator; Zellner estimator; heteroscedastic model Citations:Zbl 0129.11203 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] ANDERSON, T. W. 1971. The Statistical Analy sis of Time Series. Wiley, New York. Z. [2] BILODEAU, M. 1990. On the choice of the loss function in covariance estimation. Statist. Decisions 8 131 139. Z. · Zbl 0711.62061 [3] KARIy A, T. 1981. Bounds for the covariance matrices of Zellner’s estimator in the SUR model and the 2SAE in a heteroscedastic model. J. Amer. Statist. Assoc. 76 975 979. Z. JSTOR: · Zbl 0488.62050 · doi:10.2307/2287598 [4] KARIy A, T. and TOy OOKA, Y. 1985. Nonlinear versions of the Gauss Markov theorem and GLSE. In Multivariate Analy sis 6 345 354. North-Holland, Amsterdam. Z. · Zbl 0583.62060 [5] KHATRI, C. G. and SRIVASTAVA, M. S. 1971. On exact non-null distributions of likelihood ratio criteria for sphericity test and equality of two covariance matrices. Sankhy a Ser. A 33 201 206. Z. · Zbl 0242.62025 [6] REVANKAR, N. S. 1974. Some finite sample results in the context of two seemingly unrelated regression equations. J. Amer. Statist. Assoc. 69 187 190. Z. JSTOR: · Zbl 0281.62067 · doi:10.2307/2285521 [7] SUGIy AMA, T. 1966. On the distribution of the largest latent root and the corresponding latent vector for principal component analysis. Ann. Math. Statist. 37 995 1001. Z. · Zbl 0151.24103 · doi:10.1214/aoms/1177699378 [8] SUGIy AMA, T. 1970. Joint distribution of the extreme roots of a covariance matrix. Ann. Math. Statist. 41 655 657.Z. · Zbl 0212.51102 · doi:10.1214/aoms/1177697108 [9] TOy OOKA, Y. and KARIy A, T. 1986. An approach to upper bound problems for risks of the generalized least squares estimators. Ann. Statist. 14 679 690. Z. · Zbl 0609.62043 · doi:10.1214/aos/1176349946 [10] ZELLNER, A. 1962. An efficient method of estimating seemingly unrelated regression and tests for aggregation bias. J. Amer. Statist. Assoc. 57 348 368. Z. JSTOR: · Zbl 0113.34902 · doi:10.2307/2281644 [11] ZELLNER, A. 1963. Estimators for seemingly unrelated regressions: some exact finite sample results. J. Amer. Statist. Assoc. 58 977 992. JSTOR: · Zbl 0129.11203 · doi:10.2307/2283326 [12] KUNITACHI, TOKy O, 186 YAMAGUCHI-SHI, YAMAGUCHI-KEN JAPAN JAPAN E-MAIL: cr00055@srv.cc.hit-u.ac.jp This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.